How to use planarity efficiently: New tree-decomposition based algorithms

Abstract

We prove new structural properties for tree-decompositions of planar graphs that we use to improve upon the runtime of tree-decomposition based dynamic programming approaches for several NP-hard planar graph problems. We give for example the fastest algorithm for PLANAR DOMINATING SET of runtime 3 tw · nO(1), when we take the treewidth tw as the measure for the exponential worst case behavior. We also introduce a tree-decomposition based approach to solve non-local problems efficiently, such as PLANAR HAMILTONIAN CYCLE in runtime 6tw · nO(1). From any input tree-decomposition, we compute in time O(nm) a tree-decomposition with geometric properties, which decomposes the plane into disks, and where the graph separators form Jordan curves in the plane. © Springer-Verlag Berlin Heidelberg 2007.